THERMODYNAMIC STABILITY OF IRREVERSIBLE PROCESSES: A

GIBBS-DUHEM TYPE THEORY AND THE FOURTH LAW OF

THERMODYNAMICS

Anil A. Bhalekar1, 2, 3

and Bjarne Andresen4

1Department of Chemistry, R. T. M. Nagpur University, Amravati Road Campus, NAGPUR -

440 033, India; 2Institute for Basic Research (IBR), 35246 US 19 North # 215, Palm Harbor, FL 34684 U.S.A.;

3Thunder Energies Corporation, 150 Rainville Road, Tarpon Springs, FL 34689, U. S. A.;

4 Niels Bohr Institute, University of Copenhagen, Universitetsparken 5,

DK-2100 Copenhagen Ø, Denmark;

Keywords: Thermodynamic Stability, Gibbs-Duhem Theory of Stability,

Stability of Nonequilibrium States.

ABSTRACT

The Gibbs-Duhem theory of stability of equilibrium states has been extended to determine the

stability of irreversible processes. The basic concept of virtual displacement in the reverse

direction on the real trajectory, which is involved in the celebrated Gibbs-Duhem theory, has

been used. This establishes that all thermodynamically describable processes are

thermodynamically stable. This outcome led us to reformulate the fourth law of thermodynamics.

Moreover, our present investigations illustrate the basis of the universal inaccessibility principle

formulated earlier by one of the present authors (AAB).

INTRODUCTION

To meet the global energy demands with the limited resources at our disposal, there is

currently considerable pressure on sustainable energy management. To achieve this goal,

thermodynamic considerations enjoy a vital status among other tools. Within this approach, the

thermodynamic stability criterion plays a crucial role. To this day, we have only a very sound

thermodynamic stability theory of equilibrium states, which is known as Gibbs-Duhem stability

theory [1-4]. However, this theory applies only to a very limited extent to the real problems we

encounter day to day. This is so because the said theory is centered on equilibrium states,

whereas in real problems, one hardly deals with equilibrium situations. To remove this lacuna,

we need a thermodynamic stability theory that works entirely within a nonequilibrium

environment. The notable efforts are those of Prigogine and co-workers (see for example

reference [4] and references cited therein), in which they have developed their stability theory for

local thermodynamic equilibrium (LTE) states. In principle, this theory is not capable of

handling those vast varieties of irreversible processes not expected to lie in the domain of LTE.

Hence, a thermodynamic stability theory of a wider spectrum of applicability is required. To

meet this requirement, we have extended the Gibbs-Duhem stability theory of equilibrium states

to general nonequilibrium situations, in which no assumption like LTE is involved. Thus, it is a

comprehensive thermodynamic theory of stability.

2017 SUSTAINABLE INDUSTRIAL PROCESSING SUMMIT AND EXHIBITION Volume 2: Dodds Intl. Symp. / Energy Production

Edited by: F. Kongoli, A. Buhl, T. Turna, M. Mauntz, W. Williams, J. Rubinstein, P.L. Fuhr, M. Morales-Rodriguez, FLOGEN, 2017

109

For the sake of self-sufficiency, let us first recall some basic aspects of the Gibbs–Duhem theory

of stability of equilibrium states.

The thermodynamic description of the natural direction of irreversible evolution is [1– 4],

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

where in eqs. (1) and (2) is the entropy, is the Gibbs free energy, is the Hemholtz free

energy, is the internal energy, is the volume, is the temperature, and is the pressure.

Hence the unnatural direction of processes obviously become described as,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

These inequalities constitute the Gibbs-Duhem thermodynamic theory of stability of equilibrium

states [1– 4].

In essence, the above inequalities respectively describe the impossible changes of the state

functions or virtual displacements to a neighboring nonequilibrium states. In other words, they

are the expressions of unattainability (under the given conditions) of nonequilibrium states from

an equilibrium state. The said nonequilibrium states are the ones through which the system

naturally evolved to a final equilibrium state, as described in eqs. (1) and (2).

Let us recall that the inequalities of eqs. (1) and (2) describe the monotonic increase/decrease of

the thermodynamic functions during the given irreversible processes.

Upon close scrutiny of the above thermodynamic inequalities, a few queries get raised. They are,

1. In the above inequalities, the change of state functions can be taken equally between two

end equilibrium states, as well as between end equilibrium and nonequilibrium states. But

when the latter is the case, then how does one arrive at these state functions for

nonequilibrium states? This by itself is a thermodynamically (particularly at the times of

Clausius, Gibbs and Duhem) obvious question.

2. The preceding query crops up because we have a rigorous thermodynamic definition of

entropy, , in equilibrium only, which is given by Clausius [5],

( ) ( )

3. In equilibrium we have the following definitions of Helmholtz free energy, , and Gibbs

free energy, ,

( ) But when we consider that the above thermodynamic inequalities are between two end

states, one equilibrium and the other nonequilibrium states, then it implies that there

exists a function, , also for nonequilibrium states and hence the definitions of eq. (6)

hold true for nonequilibrium states as well.

4. It seems that the answer to the above question lies in the use of the inequality in

differential form, stated by Clausius himself without its derivation [5,6],

( ) ( )

However, Meixner, as late as in 1973, argued that the entropy function appearing in eq.

(7) must be that for the intermediate nonequilibrium states of an irreversible evolution of

a system [6].

5. In particular, the inequalities in terms of the functions and are extensively used in

chemical thermodynamics describing chemical reactions at non-vanishing rates, inter-

110

phase matter transfers, and matter diffusion (see for example [7–9]) starting with the

inequality of eq. (7).

In spite of these ambiguities associated with the above stated inequalities, the crucial principle

that is involved in the Gibbs-Duhem theory of stability of equilibrium states is the impossibility

of traversing the path in the reverse direction. This basic element we have extended to

irreversible processes. The conclusion reached is that irreversible evolutions describable

thermodynamically are all stable ones.

In doing so, we take care of not getting trapped in the types of ambiguities mentioned above. Our

investigations finally culminate into a formulation of the fourth law of thermodynamics.

The chronology of presentation is as follows. Next Section describes the thermodynamic theory

of stability of (i) irreversible processes both under traditional conditions and general evolution,

and (ii) nonequilibrium stationary states. In the following Section we formulate the fourth law of

thermodynamics. The final Section contains concluding remarks.

ENTROPY BALANCE EQUATION AND THERMODYNAMIC STABILITY OF

IRREVERSIBLE PROCESSES

Earlier we have developed a generalized phenomenological irreversible thermodynamic

theory (GPITT) [10– 20]. We in particular refer readers to the paper cited at reference [18]

wherein we have finally succeeded in identifying thermodynamic variables for spatially non-

uniform systems (also in upcoming paper [21]). Using GPITT given Gibbs relation or from any

of the other nonequilibrium thermodynamic frameworks such as classical irreversible

thermodynamics (CIT), extended irreversible thermodynamics (EIT), etc. (reviewed in reference

[22]), it is easy to recall that for an open system evolving irreversibly we have,

( )

where is the net rate of entropy change of the system, the net rate of entropy exchange is

given by,

∫( ) ( )

and the total rate of entropy production reads as,

∫ ( )

where is the entropy flux density and is the entropy source strength. Notice that eqs. (8) –

(10) are valid for a multicomponent multiphase system in nonequilibrium.

Thus, a complete and direct description of irreversibility is the rate of entropy production, eq.

(10). Hence, Prigogine writes the second law of thermodynamics in the following words [23,

24]:

Although eq. (10) is an inequality, it cannot be used to deduce the thermodynamic stability of the

corresponding irreversible processes along the lines of the Gibbs-Duhem theory of stability of

equilibrium states. This is so because it is not a complete description of the change of state

during an irreversible process. However, we notice that under certain conditions, the rate of

―Entropy of a system can change on two and only two counts, one by the

exchange mechanism and the other by its production within the system.‖

111

entropy production becomes a complete description of the change of state of a system. This

approach is described in the following Subsection.

Thermodynamic Stability of Irreversible Processes Occurring in Systems with Conditions

Imposed

Notice that once we have established an entropy function in nonequilibrium

independently using Clausius‘ inequality, it then follows that we legitimately have the following

standard relations at the local level too,

( ) where is the per unit mass Hemholtz free energy and is the per unit mass Gibbs free energy.

When and are uniform throughout a given system, we have the following standard relations

at the global level,

( ) Notice that all the functions of eqs. (11) and (12) are time dependent, because in these

expressions there appears the time dependent entropy in nonequilibrium.

Recall once again that the Gibbs-Duhem theory of stability of equilibrium states is based on the

impossibility of a virtual displacement from an equilibrium state in the reverse direction on the

trajectory of the natural evolution, to the final equilibrium state determined by eqs. (3) and (4).

We are applying exactly the same method to the irreversible segments between two end

nonequilibrium states that would determine the thermodynamic stability of irreversible

processes.

Now it is ideal to start with closed systems evolving irreversibly under certain imposed

conditions.

1. On imposing the condition of isolation on a system, all exchange mechanisms become

forbidden, eq. (8) reduces to,

(

) (

) {

( )

( )

This inequality describes the evolution towards a final equilibrium state. Also, the rate of

entropy production monotonically decreases and finally becomes zero, and as a result of

this, the entropy of the system maximizes at the final equilibrium state. Hence for its

conjugate virtual displacement, eq. (13) guarantees,

(

) (

) {

( )

( )

Therefore, the irreversible processes in the natural direction under the condition of

isolation are guaranteed to be thermodynamically stable ones. The allowed irreversible

processes are chemical reactions at non-vanishing rates, internal matter diffusion within

and across the phases of the system, and due to the existence of heat and momentum

fluxes within the system.

2. In this case, we impose on eq. (8) the condition of the constancy of and and we

obtain,

(

) (

) {

( )

( )

In arriving at the result of eq. (15) we have also used the following identity obtained

under the uniformity of and the constancy of and from the first law of

thermodynamics,

(

) (

) {

( )

( )

112

The inequality of eq. (15) describes the evolution towards a final equilibrium state. It also

demonstrates that the rate of entropy production decreases monotonically and becomes

zero at the final equilibrium state, whereby the entropy of the system maximizes at the

final equilibrium state. Therefore, for the conjugate virtual displacement eq. (15)

guarantees,

(

) (

) {

( )

( )

Thus, the isoenthalpic-isobaric evolution of a system is guaranteed to be a

thermodynamically stable one. In arriving at the above inequality, the pressure and

temperature were considered uniform throughout the system. Therefore, at constant

pressure and enthalpy, the only irreversibility that can exist is due to chemical reactions

at a non-vanishing rate and matter diffusion.

3. Let us consider a closed system evolving irreversibly under the uniform within and

across the boundaries of the closed system, with the only possible work being that due to

changes. Hence the corresponding expression of the first law of thermodynamics

is,

( ) ( )

Further assuming uniformity of within and across the boundaries of the given closed

system, we can use the standard relation of Helmholtz free energy given in eq. (12),

whose time rate variation reads as,

( ) ( )

Notice that because now we have used the entropy function for nonequilibrium states, the

function is that for a nonequilibrium state. Next on substituting eq. (18) into eq. (19)

yields,

( ) ( )

which under the condition of constant and reduces to,

(

) (

) (

) ( ) ( )

However, at constant and eq. (8) reads as,

(

) (

) (

) ( )

The second law of thermodynamics for a system at uniform temperature throughout and

across its boundaries gives,

( )

so that from eq. (23) we have,

(

)

(

) ( )

Thus, on substituting eq. (24) into eq. (21) and using eq. (22), yields the following

inequality,

(

) (

) {

( )

( )

The inequality of eq. (25) describes the evolution towards a final equilibrium state. Also,

the rate of entropy production decreases monotonically and finally becomes zero

whereby the Helmholtz free energy of the system is a minimum at the final equilibrium

state. Therefore, for its conjugate virtual displacement eq. (25) guarantees,

113

(

) (

) {

( )

( )

Thus, the evolution of a closed system at constant is guaranteed to be a

thermodynamically stable one. Arriving at the above inequalities, the uniformity of and

was assumed, such that the irreversibility on account of heat transfer within and across

the boundaries of the system, and that due to changes cannot exist anymore

because volume is also kept constant.

4. Now let us consider uniformity of and within and across the boundaries of a closed

system and also assume that only work is involved. Then the applicable expression

of the first law of thermodynamics is,

( ) ( )

Under these conditions we can use the standard expression of Gibbs free energy, , given

in eq. (12) and its time rate of change reads as,

( ) ( )

which on combining with eq. (27) yields the following expression,

( ) ( )

which at constant and takes the form,

(

) (

) (

) ( ) ( )

while at constant and eq. (8) reads as,

(

) (

) (

) ( )

Under these conditions, from eqs. (23) and (31) the rate of exchange of entropy reads as,

(

)

(

) ( )

which on substitution into eq. (30) yields,

(

) (

) {

( )

( )

The inequality of eq. (33) describes the evolution towards a final equilibrium state. Also,

the rate of entropy production decreases monotonically and finally becomes zero at

equilibrium, and whereby the Gibbs free energy of the system is minimum at the final

equilibrium state. Therefore, for its conjugate virtual displacement eq. (33) guarantees,

(

) (

) {

( )

( )

Thus, the evolution of a closed system at constant and is guaranteed to be a

thermodynamically stable one. Arriving at the above inequalities, the uniformity of and

was assumed such that the irreversibility on account of heat transfer within and across

the boundaries of the system and that due to changes cannot exist. Thus, if viscous

dissipation is also absent, then the only irreversibility that can exist is due to non-

vanishing rates of chemical reactions and internal matter diffusion.

Now we see that the inequalities of eqs. (1) and (2) are in essence the assertions about the

entropy production under the respective conditions.

Yet another deduction that follows from the conditional inequalities of eqs. (13) – (15),

(17), (25), (26), (33), and (34) is that they all guarantee that the respective evolution follows a

passage through predestined nonequilibrium states to the final equilibrium state. Also as the

114

system evolves, it passes successively from more unstable nonequilibrium states to less unstable

nonequilibrium states. This then implies that during these conditional evolutions, the

instantaneous rate of entropy production continuously decreases.

Thermodynamic Stability of Equilibrium States

In the preceding Subsection we have demonstrated the stability of irreversible evolution.

This remains true throughout the irreversible evolution. Hence, at the ‗end segment‘ of the

irreversible evolution, the virtual displacement would be considered from the final equilibrium

state to the same neighboring nonequilibrium states comprising forward motion, but in the

reverse order. Since this reverse motion constitutes an impossible transition, the stability of the

equilibrium state gets established. This is nothing else but the Gibbs-Duhem theory of stability of

equilibrium states, and its specific descriptions get generated on integration of inequalities of this

Section between the end equilibrium state and the neighboring nonequilibrium states. They are

none else than the traditional ones given in eqs. (1) and (3). Correspondingly, if we consider the

differential rates of this Section as the ones between the end equilibrium state and the

infinitesimally away nonequilibrium state, then they are nothing else than those given in eqs. (2)

and (4). This establishes the self-consistency of the present analysis of stability of irreversible

processes.

Thermodynamic Stability of Nonequilibrium Stationary States (NSS)

Lavenda has claimed that it is not possible to thermodynamically assign maximum

entropy to a nonequilibrium stationary state (NSS) (see Chapter 4 in [25]). However, it is

possible to establish thermodynamic stability of NSS. Let us first elaborate the above assertion of

Lavenda.

Thermodynamically at nonequilibrium stationary state we have from eq. (8),

( )

where on the right-hand side of the second equality, the first term is the rate of exchange of

entropy, the second term quantifies the rate of production of entropy, and the superscript

denotes nonequilibrium stationary state. Then the Taylor expansion of entropy about its value

at NSS on a path that led the system to achieve the given NSS yields,

( ) ( )( )

( )( ) ( ) ( )

Notice that it is not possible to arrive at ( ) without imposition of some suitable

conditions, but then under those conditions it would not be possible to attain a NSS. That is, one

cannot assert that the entropy at NSS is maximum or in other words, the entropy about NSS is

not obtained as a convex function. Hence on differentiating eq. (36) w.r.t. time we get,

( )

( )

( ) ( )

Now eq. (37) needs to be compared with eq. (8) and we see in eq. (8) that the rate of entropy

change is determined by the flow ( ⁄ ) and the non-flow ( ⁄ ) terms.

Hence, eq. (37) needs to be composed of flow and non-flow terms. Therefore, eq. (37) gets

expressed as,

115

(

( )

)

( ( )

)

( ( )

)

( ( )

)

( )

where the first set of terms on the right-hand side of eq. (38) must measure the rate of exchange

of entropy,

(

( )

)

( ( )

)

( )

and the second set of terms of eq. (38) must measure the rate of production of entropy,

(

( )

)

( ( )

)

( )

Because of the existence of a contribution from flow terms, we have no definite sign in eq. (38).

Hence on the attainment of an NSS the entropy is not maximized in general.

As far as the thermodynamic stability of NSS is concerned, one needs to recall that when

the constraints on the system are such that it finally attains a NSS, which is thermodynamically

described by eq. (8) it finally reduces to eq. (35). This motion is unidirectional from

nonequilibrium to the final NSS. Therefore, the reverse motion from the said NSS to any of the

nonequilibrium states from which the said NSS was achieved is impossible. That is, if from eq.

(36) for the forward motion we have ( ) , then it is impossible to have a motion

commensurate with ( ) . This can also be understood by the fact that the exchange

of entropy contribution to ( ) would change sign on reversing the motion, while

simultaneously the entropy production contribution cannot be made negative. Hence, the so

called virtual displacement would carry the system on paper to some other succession of

nonequilibrium states, but that cannot be taken as the reverse motion. In this way, we have

proven the impossibility of virtual displacement described by ( ) and hence all NSS

are thermodynamically stable states. This is the Gibbs-Duhem theory specialized for NSS.

Thermodynamic Stability of a General Irreversible Evolution

So far, we have demonstrated the thermodynamic stability of (i) conditional irreversible

evolution of closed systems, (ii) equilibrium states, and (iii) nonequilibrium stationary states

(NSS). However, it is possible to establish thermodynamic stability of a general irreversible

evolution as follows.

Consider the general statement of eq. (8), which is valid for open systems too. It

describes an irreversible motion and correspondingly establishes forward direction. This is

demonstrated by proving that this description cannot be reversed. For a system to have a reversed

motion all the terms of eq. (8) must be made to reverse their signs. However, ⁄

cannot be assigned a negative sign, because that will shatter the very definition of a

nonequilibrium state. That means on changing the sign of ⁄ the system would attain some

other nonequilibrium state, which cannot be called a reverse motion. Alternatively, suppose that

in a given irreversible motion ⁄ irrespective of the sign of ⁄ , then the

envisaged reverse motion described by ⁄ is impossible because the rate of entropy

production cannot change its sign. It thus establishes that eq. (8) indeed describes the forward

motion that cannot be reversed. Thus, in general no irreversible motion describable by eq. (8)

can be reversed. That means the virtual displacement in the reverse direction of a forward

motion described by eq. (8) cannot be realized, and every irreversible motion describable by eq.

(8) is obtained as a stable one. This also demonstrates the basis of the statement of the Universal

116

Inaccessibility Principle (UIP) [10, 12] formulated earlier by one of the present authors (AAB),

which reads as,

THE FOURTH LAW OF THERMODYNAMICS

The discussions and conclusions presented in the preceding Section lead us to formulate a

new law of thermodynamics. That we term as the fourth law of thermodynamics that reads as,

In this connection we also recall that earlier, the fourth law of thermodynamics was proposed

(for example, recently Kamal has described earlier attempts of formulation of the fourth law of

thermodynamics [26]), but none of these proposals could advance beyond the proposal stage.

However, the proposal originating from the works of Glansdorff and Prigogine on the stability of

the so-called local equilibrium states merits mention herein [4, 27, 28]. They have proposed the

following condition of stability of local equilibrium states,

( ) ( )

which seemingly resembles Lyapunov‘s direct (second) method of stability of motion [29– 40];

Glansdorff and Prigogine claim the same. That is, has been assumed to serve as the

thermodynamic Lyapunov function. The thermodynamic basis of the second inequality in eq.

(41) has been questioned [25, 40], although the very assumption of local equilibrium allows one

to consider the local entropy a convex upward function, so that the first inequality in eq. (41) is

legitimized. It is pertinent to cite a very recent example wherein Sieniutycz and Kuran (private

communication) have shown that far from equilibrium fails to serve as a Lyapunov

function. Hence, the conditions of eq. (41) are not general and hence cannot be taken as the

fourth law of thermodynamics.

However, in Lyapunov‘s theory of stability of motion an act of sufficiently small

perturbation from the real trajectory is involved, hence we have a perturbed trajectory distinctly

different from the unperturbed trajectory. In this way, the identified Lyapunov function has to be

a function of perturbation coordinates, not dependent only on the thermodynamic variables on

the real trajectory. In view of this requirement, may perhaps be considered a Lyapunov

function on the real trajectory, with the virtual displacement coordinates as the perturbation

coordinate. But then the difficulty is that the virtual displacements, in principle, would carry the

system on the real trajectory in the reverse direction. If so, how can we define corresponding

perturbation coordinates distinctly different from the thermodynamic variables on the real

trajectory? Yet another difficulty is that on a real trajectory, by the assumption of local

thermodynamic equilibrium, each point is a local equilibrium state; or in other words, the real

trajectory is made up of a succession of local equilibrium states. The perturbation, in principle,

needs to carry the system to a neighboring nonequilibrium state, which is not a local equilibrium

state. Otherwise, we will be guilty of questioning the very proposal of local equilibrium

assumption. Therefore, the Lyapunov method of stability cannot be applied if it is a case of

𝑑𝑆

𝑑𝑡

𝑑𝑒𝑆

𝑑𝑡

𝑑𝑖𝑆

𝑑𝑡

𝑑𝑖𝑆

𝑑𝑡

―The irreversible processes describable by,

cannot be reversed, hence they all constitute thermodynamically stable processes.‖

―In every neighborhood of every equilibrium and nonequilibrium state of a system there

exist states (both equilibrium and nonequilibrium ones), which remain unattainable by

reversible or spontaneous paths.‖

117

virtual displacement in the reverse direction on a path consisting of a succession of local

equilibrium states. Hence, the Glansdorff-Prigogine theory of stability of local equilibrium states

cannot be considered as a thermodynamic Lyapunov theory. Thus, if it is a case of virtual

displacement on the real trajectory in reverse direction, then there is no room to call a

Lyapunov function and then use its time derivative. Further, as stated above, in addition to these

difficulties, no thermodynamic basis of the second inequality in eq. (41) could be established*.

Of course, Landsberg [41] did suggest that it may be called a new ―law‖ that appears to be the

fourth law of thermodynamics. Unfortunately, his optimism did not stand the test of time.

In contrast, our formulation of thermodynamic stability theory and the fourth law of

thermodynamics are based purely on a Gibbs-Duhem type stability theory for irreversible

processes described in the preceding Sections— that is, it is based on the concept of virtual

displacement in the reverse direction on the real trajectory.

CONCLUDING REMARKS

A long pending demand to have a thermodynamic theory of stability of irreversible

processes has been met in this paper. It is in essence an extension of Gibbs-Duhem theory of

stability of equilibrium states to systems evolving irreversibly. With the advent of this theory,

almost all branches of science would benefit, such as industrial processes, energy optimization

management, biophysics, processes affecting/maintaining ecology, physico-chemical processes,

etc.

We notice that the inequalities of eqs. (1) – (4) can now be understood as describing

natural and unnatural changes of closed systems (i) between end equilibrium states, (ii) between

equilibrium and nonequilibrium states, and (iii) between end nonequilibrium states. This is easily

demonstrated by appropriately integrating the inequalities of eqs. (13), (25), and (33) that yield

the inequalities of eq. (1) for the categories (i), (ii), and (iii). Similarly, appropriately integrating

the inequalities of eqs. (14), (26), and (34), yields the inequalities of eq. (3) for the categories (i),

(ii), and (iii). On the other hand, for infinitesimally away states when we drop the dt from the

denominator from the inequalities of eqs. (13), (14), (25), (26), (33), and (34), the inequalities of

eqs. (2) and (4) are obtained for the categories (i), (ii), and (iii).

The striking outcome is that the irreversible processes describable by eq. (8) all become

established as thermodynamically stable ones. On the face of it, it looks like a strange outcome,

but we need to keep in mind that basically thermodynamics is not a constitutive theory, it only

provides broad guidelines, and for example eq. (8) describes irreversible evolution of a system

with its behavioral contents, but not the constitutive rates of the other properties of the system.

Of course, on elaborating the rate of change of entropy in terms of the thermodynamic variables

and then on supplying constitutive relations of the rates of change of thermodynamic variables,

the corresponding thermodynamic predictions would be generated. To understand this, let us

recall that in chemical thermodynamics, on providing the constitutive equations, say ideal gas

law or van der Waals equation, the corresponding thermodynamic predictions get generated.

Similarly, in the case of natural evolution of an isolated system, for example, if the irreversibility

is on account of a simple bimolecular reaction at a non-vanishing rate, the rate of change of

entropy is given by [8, 23, 24],

( )

* Of course, even if at the global level under certain global conditions, such as isolation of the system where we have

, this does not guarantee, in general, at a local level because each local pocket behaves as a tiny

open system. 118

where is the chemical affinity and is the extent of advancement of the chemical reaction.

Whereas from chemical kinetics [42] the rate of the chemical reaction, say

( ) with the forward rate constant , is given by,

( ⁄ )

( ⁄ ) ( )

where and are the concentrations of the reactants A and B respectively, is the Arrhenius

pre-exponential factor, is the Arrhenius energy of activation of the forward chemical reaction,

is the Boltzmann constant, is Planck‘s constant, is the Gibbs free energy of activation

in going from reactants to activated complex ( ) at the top of the Gibbs free energy barrier of the

chemical reaction, and R is the universal gas constant.

In eq. (44), the first expression of the rate of reaction is based on laboratory experiments,

the second expression is the Arrhenius theoretical version [43], and the third expression is also a

theoretical one based on the Eyring‘s theory of bimolecular reaction rates [43– 49]. Thus, there

are three possibilities to evaluate the rate of entropy change of eq. (42). In this way we will have

corresponding thermodynamic predictions, as well as access to the respective control parameters.

The present study tells us that, to achieve stability, one needs to maintain inaccessibility

in the reverse direction on the trajectory of a given motion. Thus, we learn that if the conditions

of irreversible evolution are strictly and meticulously maintained, in reality the thermodynamic

stability would be maintained. Hence, the challenge before us is to maintain those conditions

rigorously during an irreversible process under investigation. However, the stability/instability

obtained from non-thermodynamic means constitutes a constitutive theoretical investigation;

indeed, they have their own practical advantages too. The distinction between the two is akin to

the distinction between thermodynamics and statistical mechanics. Moreover, in order to retain

the full thermodynamic character of the present theory, we have not incorporated the aspects

dealing with the hydrodynamic stability, as Glansdorff and Prigogine have done in their

proposal.

The present discussion also sheds light on the effect of the fluctuations about equilibrium

and nonequilibrium stationary states on thermodynamic properties of a given system.

Statistically, all possible fluctuations leading a system to neighboring nonequilibrium states are

counted, but only overwhelmingly dominating fluctuations are those, which produce the virtual

displacements in the reverse direction on the trajectory described by eq. (8) in the forward

direction.

Though the present theory does not need the Lyapunov type of analysis, one would still

be interested to know if there can be a thermodynamic tool akin to Lyapunov‘s theory of stability

of motion [29– 40]. Yes, one of the present authors (AAB) has already developed a

comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP) [50– 54].

In CTTSIP we apply all the steps of Lyapunov‘s direct method of stability of motion to the

defined thermodynamic Lyapunov function via the entropy production function on real and

perturbed trajectories. This approach we are summarizing in the following paper [55].

ACKNOWLEDGEMENTS

This work was started during the Emeritus Scientist (CSIR, New Delhi) tenure (January 2008 –

December 2010) of one of the present authors (AAB) but only got materialized now. The

119

financial support in Emeritus Scientist Scheme from CSIR, New Delhi (No. 21(0695)/07/EMR-

II) is gratefully acknowledged by AAB.

REFERENCES

[1] H. A. Bumstead and R. G. V. Name, eds., The Scientific Papers of J. Willard Gibbs, vol.

I. Thermodynamics, 1906, Longmas, Green and Company, London and Bombay.

[2] F. G. Donnan and A. Haas, eds., A Commentary on the Scientific Writings of J. Willard

Gibbs, vol. I, 1936, Yale University Press, London, New Haven.

[3] P. Needham, Commentary on the Principles of Thermodynamics by Pierre Duhem, 2011,

Springer, Dordrecht, New York.

[4] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and

Fluctuations, 1971, Wiley, New York.

[5] W. H. Cropper, ―Rudolf Clausius and the road to entropy,‖ Am. J. Phys., 54 (1986), 1068

– 1074.

[6] J. Meixner, ―The entropy problem in thermodynamics of processes,‖ Rheol. Acta., 12(3)

(1973), 465 – 467.

[7] K. G. Denbigh, The Principles of Chemical Equilibrium, 1973, Cambridge University

Press, Cambridge, UK.

[8] K. G. Denbigh, The Thermodynamics of the Steady State, 1958, Methuen & Company,

London.

[9] S. Glasstone, Thermodynamics for Chemists, 1967, D. Van Nostrand, New Jersey.

[10] A. A. Bhalekar, ―On the generalized phenomenological irreversible thermodynamic

theory (GPITT),‖ J. Math. Chem., 5(2) (1990), 187 – 196.

[11] A. A. Bhalekar, ―Measure of dissipation in the framework of generalized

phenomenological irreversible thermodynamic theory (GPITT),‖ Proc. Int. Symp. on

Efficiency, Costs, Optimization and Simulation of Energy Systems (ECOS‘92), Zaragoza,

Spain, June 15-18, 1992, A. Valero and G. Tsatsaronis, eds.; Empresa Nacional de

Electricidad, Spain and Amer. Soc. Mech. Eng., pp. 121 – 128.

[12] A. A. Bhalekar, ―Universal inaccessibility principle,‖ Pramana - J. Phys., 50(4) (1998),

281 – 294.

[13] A. A. Bhalekar, ―On the generalized zeroth law of thermodynamics,‖ Indian J. Phys.,

74B( 2) (2000) 153 – 157.

[14] A. A. Bhalekar, ―On the time dependent entropy vis-à-vis Clausius‘ inequality: Some of

the aspects pertaining to the global and the local levels of thermodynamic

considerations,‖ Asian J. Chem., 12(2) (2000), 417 – 427.

[15] A. A. Bhalekar, ―On the irreversible thermodynamic framework for closed systems

consisting of chemically reactive components,‖ Asian J. Chem., 12(2) (2000), 433 – 444.

[16] A. A. Bhalekar and B. Andresen, ―On the nonequilibrium thermodynamic roots of the

additional facets of chemical interaction,‖ in Recent Advances in Thermodynamic

Research Including Nonequilibrium Thermodynamics, Proceedings of 3rd

National

Conference on Thermodynamics of Chemical and Biological Systems (NCTCBS-2008),

G. S. Natarajan, A. A. Bhalekar, S. S. Dhondge, and H. D. Juneja, eds., (Department of

Chemistry), 2008, pp. 53 – 62, R. T. M. Nagpur University, Nagpur, Funded by CSIR,

New Delhi, October 16– 17.

[17] A. A. Bhalekar, ―The dictates and the range of applicability of the laws of

thermodynamics for developing an irreversible thermodynamical framework,‖ Indian J.

Phys., 76B (2002), 715 – 721.

120

[18] A. A. Bhalekar, ―Irreversible thermodynamic framework using compatible equations

from thermodynamics and fluid dynamics. A second route to generalized

phenomenological irreversible thermodynamic theory (GPITT),‖ Bull. Cal. Math. Soc.,

94(2) (2002), 209 – 224.

[19] A. A. Bhalekar, ―The universe of operations of thermodynamics vis-à-vis Boltzmann

integro-differential equation,‖ Indian J. Phys., 77B (2003), 391 – 397.

[20] A. A. Bhalekar, ―Thermodynamic insight of irreversibility,‖ Proceedings of the Third

International Conference on Lie-Admissible Treatment of Irreversible Processes

(ICLATIP - 3), Dhulikhel, Kavre, Nepal, 2011, C. Corda, ed., pp. 135 – 162, R. M.

Santilli Foundation, USA and Kathmandu University.

[21] A. A. Bhalekar and B. Andresen, ―A comprehensive formulation of generalized

phenomenological irreversible thermodynamic theory (GPITT),‖ (to appear).

[22] G. Lebon, D. Jou and J. Casas-Vázquez, Understanding of Non-equilibrium

Thermodynamics. Foundations, Applications and Frontiers, 2008, Springer, Berlin.

[23] I. Prigogine and R. Defay, Chemical Thermodynamics, 1954, Longmans Green, London,

Translated by D. H. Everett.

[24] I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 1967, John

Wiley-Interscience, New York.

[25] B. H. Lavenda, Thermodynamics of Irreversible Processes, 1978, Macmillan Press,

London.

[26] S. A. Kamal, ―The fourth law of thermodynamics,‖ The Pakistan Institute of Physics

Conference, Lahore, March 2011, 1 – 5,. Paper # PIPC-11-25.

[27] P. Glansdorff and I. Prigogine, ―Non-Equilibrium stability theory,‖ Physica, 46 (1970),

344 – 366.

[28] P. Glansdorff, G. Nicolis, and I. Prigogine, ―The thermodynamic stability theory of non-

equilibrium states,‖ Proc. Nat. Acad. Sci. USA, 71 (1974), 197–199.

[29] I. G. Malkin, ―Theory of stability of motion,‖ in ACE-tr-3352 Physics and Mathematics,

1952, US Atomic Energy Commission, Washington, New York, Moscow, Leningrad.

[30] N. G. Chetayev, The Stability of Motion. 1961, Pergamon Press, Oxford, M. Nadler,

Transl.

[31] I. Z. Shtokalo, Linear Differential Equations with Variable Coefficients. Criteria of

Stability and Instability of Their Solutions, 1961, Hindustan Publishing, Delhi, India.

[32] I. G. Malkin, Stability and Dynamic Systems, 1962, American Mathematical Society,

Providence, Rhode Island.

[33] Reference 32, pp. 291– 297.

[34] W. Hahn, Theory and Applications of Lyapunov’s Direct Method. 1963, Prentice-Hall,

Englewood Cliffs, New Jersey.

[35] J. P. LaSalle and S. Lefschetz, eds., Nonlinear Differential Equations and Nonlinear

Mechanics. 1963, Academic Press, New York.

[36] L. Elsgolts, Differential Equations and the Calculus of Variations, 1970, Mir

Publications, Moscow, G. Yankuvsky, Transl.

[37] H. H. E. Leipholz, Stability Theory: An Introduction to the Stability of Dynamic Systems

and Rigid Bodies, 1987, B. G. Teubner/John Wiley, Stuttgart/Chichester.

[38] J. P. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method with Applications,

1961, Academic Press, New York.

[39] D. A. Sànchez, Ordinary Differential Equations and Stability Theory. An Introduction,

1979, Dover, New York.

121

[40] J. Keizer and R. F. Fox, ―Qualms regarding the range of validity of the Glansdorff-

Prigogine criterion for stability of non-equilibrium states,‖ Proc. Nat. Acad. Sci. USA, 71 (1974), 192 – 196, 2919.

[41] P. T. Landsberg, ―The fourth law of thermodynamics,‖ Nature, 238 (1972), 229 – 231.

[42] K. J. Laidler, Chemical Kinetics, 1967, Tata McGraw-Hill, New Delhi.

[43] R. Wayne, The Theory of the Kinetics of Elementary Gas Phase Reactions, in Comprehensive Chemical Kinetics, 1969, Eds. C. H. Bamford and C. F. H Tipper, Vol. 2, Elsevier, New York, Ch. 3, pp. 189 – 301.

[44] H. Eyring, ―The activated complex in chemical reactions,‖ J. Chem. Phys., 3 (1935), 107

– 115.

[45] S. Glasstone, K. J. Laidler, and H. Eyring, Theory of Rate Processes, 1941, First ed., McGraw-Hill, New York.

[46] M. G. Evans and M. Polanyi, ―Some applications of the transition state method to the calculation of reaction velocities, especially in solution,‖ Trans. Faraday Soc., 31 (1935), 875 – 894.

[47] K. J. Laidler and J. C. Polanyi, ―Theories of the Kinetics of Bimolecular Reactions‖, Progress in Reaction Kinetics, Vol. 3, 1965, Pergamon Press, London, Ch. 1, pp. 1 – 61.

[48] C. L. Arnot, ―Activated complex theory of bimolecular gas reactions,‖ J. Chem. Educ., 49

(1972), 480 – 482.

[49] A. A. Bhalekar ―The transition state theory of bimolecular reaction rates via the Bodenstein steady state for activated complexes,‖ CACAA, 4 (2015), 309 – 340.

[50] A. A. Bhalekar, ―On a comprehensive thermodynamic theory of stability of irreversible processes: A brief introduction,‖ Far East J. Appl. Math., 5 (2001), 199 – 210.

[51] A. A. Bhalekar, ―Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). I. The details of a new theory based on Lyapunov‘s direct method of stability of motion and the second law of thermodynamics,‖ Far East J. Appl. Math., 5

(2001), 381 – 396.

[52] A. A. Bhalekar, ―Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP). II. A study of thermodynamic stability of equilibrium and nonequilibrium stationary states,‖ Far East J. Appl. Math., 5 (2001), 397 – 416.

[53] A. A. Bhalekar, ―The generalized set-up of comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP) and a few illustrative applications,‖ J. Indian Chem. Soc., 81 (2004), 1119 – 1126.

[54] V. M. Tangde, S. G. Rawat, and A. A. Bhalekar, ―Comprehensive thermodynamic theory of stability of irreversible processes (CTTSIP): The set-up for autonomous systems and application,‖ Int. J. Eng. Tech. Res., 3 (2015), 182 – 191.

[55] A. A. Bhalekar and V.M. Tangde, ―Thermodynamic stability of irreversible processes based on Lyapunov function analysis,‖ in Volume 2: Dodds Intl. Symp. / Energy Production, 2017 Sustainable Industrial Processing Summit & Exhibition (SIPS 2017), F. Kongoli, A. Buhl, T. Turna, M. Mauntz, W. Williams, J. Rubinstein, P.L. Fuhr, M. Morales-Rodriguez, eds., 2017, pp. 153-173, FLOGEN Star Outreach, Montreal, October

22-26.

122